Cellular multiuser multiple-input multiple-output (MIMO) systems are under continuous development for future wireless communication. One topic is the maximization of cell-edge user rates in interference-limited cellular systems. Inter-cell interference (ICI) is the system bottleneck for current cellular networks and many existing methods have tried to mitigate it using different techniques. Users at cell-edges mostly suffer from strong interference of neighbouring base stations and this directly reflects in lower achievable rates.
FIG. 11A shows a schematic illustration of a two-cell MISO (multiple-input single-output) system model for a MISO linear precoding. It shows two base stations BS1, BS2 covering cell 1 and cell 2 and two wireless devices UE1, UE2 (user equipment 1, user equipment 2). Further, FIG. 11B shows the direct links (useful signal) and interfering links (interference) by schematic illustration of the channels between the base stations BS1, BS2 and the wireless devices UE1, UE2, when base station 1 uses beamformer b1 and base station 2 uses beamformer b2.
In other words, a two-cell MISO scenario with a single user per cell is considered as shown in FIG. 11A, where each BS has M antennas and each receiver at the cell-edge has one antenna. This system model could be also considered as a two-user MISO interference channel (IC), as shown in FIG. 12, where a user in the context of ICs denotes a transmitter/receiver pair. Data symbol si˜(0, 1) (i=1, 2) is linearly precoded by precoder piεM×1 and transmitted over the channel hiiεM×1 to be received by user i. Due to the system nature, it will also be transmitted on channel hjiεM×1, j≠i, and received by user j as undesired interference. The transmitted signal is subject to a transmit power constraint E[∥pisi∥22]=Etxi. At the receiver side, the obtained signal is perturbed by noise ni˜(0, σi2). Writing the obtained estimates of the data symbols gives:ŝ1h11Tp1s1+h12Tp2s2+n1 ŝ2h22Tp2s2+h21Tp1s1+n2  (1)where (●)T denotes transposition.
The most common measure that captures system performance is given by the achievable sum rate C:
                              C          =                                    ∑                              i                =                1                            2                        ⁢                                                  ⁢                                          log                2                            ⁡                              (                                  1                  +                                      SINR                    i                                                  )                                                    ,                            (        2        )            where SINR1 and SINR2 are the received signal-to-interference noise ratios of receiver 1 and 2, respectively:
                                          SINR            1                    =                                    |                                                h                  11                  T                                ⁢                                  p                  1                                            ⁢                              |                2                                                    |                                                h                  12                  T                                ⁢                                  p                  2                                            ⁢                              |                2                            ⁢                              +                                  σ                  1                  2                                                                    ⁢                                  ⁢                              SINR            2                    =                                                    |                                                      h                    22                    T                                    ⁢                                      p                    2                                                  ⁢                                  |                  2                                                            |                                                      h                    21                    T                                    ⁢                                      p                    1                                                  ⁢                                  |                  2                                ⁢                                  +                                      σ                    2                    2                                                                        .                                              (        3        )            
In the following, the indices i and j where iε{1, 2} and j≠i.
A known sum rate maximization approach is the distributed interference pricing algorithm described in “D. A. Schmidt, C. Shir, R. A. Berry, M. Honig and W. Utschick, ‘Distributed Resource Allocation Schemes’, IEEE Signal Processing Magazine, September 2009, pp. 53-63”. This iterative method starts with each receiver announcing an interference price to interfering base stations (BSs), assuming cells with single users. In practice, each receiver feeds back these prices to its corresponding base station (BS) and the latter communicates them to other BSs; thus, BS cooperation is necessitated. The interference price of each receiver depends on the initial beamformers of the interfering BSs. Then, each BS separately performs a maximization of its corresponding user rate, taking into account the interference prices announced by other receivers and therefore it is a distributed approach. Therefore, it can be thought of as an egoistic approach subject to some penalty paid when causing interference to other users. The maximization results in new beamformers. Next, receivers update their interference prices and new beamformers are again calculated according to the updated interference prices. The process repeats until convergence.
Different approaches for mitigating interference in interference-limited systems have been considered. So far, the best approaches that have been proposed are distributed approaches, in which each transmitter tries to maximize its own rate taking into account interference prices announced by receivers in the system (C. Shi, R. A. Berry and M. Honig, “Distributed Interference Pricing with MISO Channels”, in Proc. 46th Annual Allerton Conference 2008, Urbana-Champaign, Ill., September 2008, pp. 539-546 and D. A. Schmidt, C. Shir, R. A. Berry, M. Honig and W. Utschick, “Distributed Resource Allocation Schemes”, IEEE Signal Processing Magazine, September 2009, pp. 53-63). Formally, interference price it represents the marginal decrease in the rate of receiver i following a marginal increase in interference caused by transmitter j, and is defined as:
                                          π            i                    =                      -                                          ∂                                  u                  i                                                            ∂                                  I                  i                                                                    ,                            (        4        )            where ui=log2(1+SINRi) is the rate of receiver i and Ii=|hijTpj|2 is the interference power present at the receiver i (see equation 3).
Given fixed interference prices, each BS i solves the following problem:
                                          p                          i              ,              opt                                =                                                                      argmax                                      p                    i                                                  ⁢                                                                  ⁢                                  u                  i                                            -                                                π                  j                                ⁢                                                                                                                        h                        ji                        T                                            ⁢                                              p                        i                                                                                                  2                                ⁢                                                                  ⁢                                  s                  .                  t                  .                                                                          ⁢                                      p                    i                    H                                                  ⁢                                  p                  i                                                      =                          E                              tx                i                                                    ,                  i          =          1                ,        2        ,                            (        5        )            where (●)H denotes conjugate transposition. The objective function of each BS can be viewed as its achievable rate minus the cost of interference it generates to other users. It is an egoistic approach that takes into consideration the penalty paid when causing interference to other users. To implement this algorithm, each receiver should announce an interference price to every interfering BS. In practice, each receiver feeds back these prices to its corresponding base station (BS) and the latter communicates them to other BSs; thus, BS cooperation is necessitated. Given these interference prices, each BS calculates its best precoder. The algorithm iteratively updates the precoders and interference prices until convergence is reached. To calculate interference prices, every receiver necessitates the knowledge of useful and interfering signal power. No precoder knowledge at the receiver side is necessary. To calculate optimal precoders, every BS i necessitates the knowledge of the channel gains hki, k=1, 2.
FIG. 11A shows an example for a target configuration in a downlink transmission in a multicell COMP/MIMO system (coordinated multipoint/multiple-input multiple-output system) as described by the distributed interference pricing algorithm. Cooperative beamforming can be used in order to maximize cell-edge user rates, but the closed form solution does not exist and a signaling overhead results. FIG. 13 schematically illustrates the distributed interference pricing algorithm between two base stations. First an announcement of interference prices pi penalties is transmitted by the receivers (wireless devices). The interference price represents the marginal decrease in rate for a marginal increase in interference. Then, the base stations iteratively maximize their own rate taking into account to announce prices pi, which is an egoistic approach. For example, base station 1 solves the following equation:
            log      ⁡              (                  1          +                      SINR            1                          )                    ︸              Rate        ⁢                                  ⁢        user        ⁢                                  ⁢        1              +            f      ⁡              (                  p          2                )                    ︸      penalty      
The cooperative beamforming jointly computes b1 and b2 to maximize C, but closed form solutions of beamformers do not exist and it is questionable what should be signaled and at which expense (overhead).
With this approach a signaling phase before each iteration may be used. It optimizes iteratively cell-edge user rates subject to interference penalties from neighboring cells. The interference penalties are signaled over the air link and then exchanged over the backhaul (per iteration).
FIG. 14 illustrates the initial signaling and the signaling phase of each iteration by a schematic illustration of the channels between two base stations and two wireless devices UE1, UE2. Base station 1 finds b1 which maximizes the following equation:
            log      ⁡              (                  1          +                      SINR            1                          )                            ︸                  Rate          ⁢                                          ⁢          user          ⁢                                          ⁢          1                            (                              approximated            ⁢                                                  ⁢            by                    ⁢                                          |                                    h              11              T                        ⁢                          b              1                                ⁢                      |            2                          )              -                    p        2            |                        h          21          T                ⁢                  b          1                    ⁢              |        2                    ︸              penalty        ⁢                                  ⁢        due        ⁢                                  ⁢        to        ⁢                                  ⁢        interfrence            
This approach has a slow convergence due to the egoistic approach, needs high computational power, is non-adaptive to fast varying systems, needs additional processing at the receiver side and additional communication via the air link (wireless link).